How Should We Improve the Ray-tracing Method?

The possibility is discussed to improve the ray approximation up to an exact representation of a wave field by the Feynman–Kac probabilistic formula (this formula gives an exact solution of the Helmholtz equation in the form of the expectation of a certain functional on the space of Brownian random walks). Some examples illustrate an application of the solutions obtained to diffraction problems. It would not be an exaggeration to claim that a large (if not a major) part of studies in the theory of wave propagation is related somehow to the ray-tracing method. The contribution of Babich and Buldyrev to the creation of this method was quite substantial. The ray-tracing method is based on the universal “divide and conquer” principle, which is revealed, in the present case, in splitting a difficult problem into two simpler ones. For instance, a solution of the Helmholtz equation (1) ∇φ+ pk(x)φ = 0, p = const, is sought in the form (2) φ(x) = u(x)e; i.e., an unknown function φ(x) is expressed in terms of two unknown functions u(x) and s(x). It is easily seen that the new unknowns must satisfy the equation ∇u+ 2ip∇s∇u+ (ip∇s+ pk − p(∇s))u = 0, which does not seem to be simpler than (1) but, in recompense, can be split in the eikonal equation (3) (∇s) = k and the transport equation (4) 1 2pi ∇u+A∇u+Bu = 0 with the coefficients A = ∇s, B = 1 2 ∇s, determined by (3). The procedure described above is feasible because the eikonal equation (3) can be solved exactly with the use of standard methods of Hamiltonian mechanics, which determine s(x) uniquely on a certain (maybe, many-sheeted) domain G̃ that covers the physical domain G. As to the transport equation (4), in distinction to the Helmholtz 2010 Mathematics Subject Classification. Primary 81Q30.